Editing 2DStructure/ToroidalCoordinates (section)

===Identifying Limits of Integration===

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<tr><th align="center"><font size="+1">Figure 2</font></th></tr>
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[[File:TCoordsE.gif|300px|Diagram of Torus and Toroidal Coordinates]]
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The animation shown here in Figure 2 builds upon the configuration displayed in [[#THH12Figure4|our Figure 1, above]].  It shows a meridional cross-section through the selected (pink) uniform-density, toroidal mass distribution, whose geometric properties are fully determined by specifying values for <math>~\varpi_t</math> and <math>~r_t</math>.  (For the example illustrated in Figure 2, we have specified the same values used by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Hur&eacute; &amp; Hersant (2012)] to construct their [[#THH12Figure4|Figure 4, as reprinted above]]; namely, <math>~\varpi_t = 3/4</math> and <math>~r_t = 1/4</math>.)  Throughout the Figure 2 animation sequence, these two parameters have been fixed &#8212; thereby fixing the properties of the (pink) torus &#8212; and, in addition, we have fixed the location of the origin of a toroidal coordinate system &#8212; identified by the red-filled circular dot.  We explicitly associate this coordinate-system origin with the (cylindrical) coordinates of the point in space at which we choose to evaluate the gravitational potential, namely, <math>~(R_*, Z_*) = (a, Z_0)</math>. [For illustration purposes, in Figure 2 we have set <math>~(a, Z_0) = (1/3, 3/4)</math>.]

While the values of the four primary model parameters <math>~(a, Z_0,  \varpi_t, r_t)</math> are held fixed, Figure 2 depicts in a quantitatively precise manner how the size of a <math>~\xi_1</math>= constant toroidal surface (the off-center circle traced by the sequence of black dots) varies as the value of the radial coordinate, <math>~\xi_1</math>, is varied.  In each frame of the animation sequence, the value of <math>~\xi_1</math> that was used to define the (black) <math>\xi_1</math>-circle is printed in the lower-right corner of the image; additional quantitative details associated with each animation frame can be obtained from the [[Appendix/Ramblings/ToroidalCoordinates#Example2|table titled, "Example 2" in our accompanying notes]].


As the value of <math>~\xi_1</math> is varied from large values (small black circles) to smaller values (larger black circles), there is a maximum value, <math>~\xi_1|_\mathrm{max}</math>, at which the <math>\xi_1</math>-circle first makes contact with the (pink) equatorial-plane torus, and there is a minimum value, <math>~\xi_1|_\mathrm{min}</math>, at which it makes its final contact.  These are the limiting values of the toroidal radial coordinate to be used in the integration that produces <math>~q_0</math>.  At all values within the parameter range,
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<math>~\xi_1|_\mathrm{max} > \xi_1 > ~\xi_1|_\mathrm{min} \, ,</math>
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the <math>\xi_1</math>-circle intersects the surface of the torus in two locations, defined by two different values of the associated angular coordinate, <math>~\xi_2</math>.  In each frame of the animation, points of intersection are marked with small yellow diamonds; the coordinates of these points of intersection are listed in the table associated with [[Appendix/Ramblings/ToroidalCoordinates#Example2|example 2, in our accompanying notes]].  For each relevant value of <math>~\xi_1</math>, these are the limiting values of the toroidal angular coordinate to be used in the integration that produces <math>~q_0</math>.  It should be realized that, ''at'' the first and final points of contact, the two values of <math>~\xi_2</math> are degenerate.  
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Next we derive the mathematical relations that give the values of <math>~\xi_1|_\mathrm{max}</math> and <math>~\xi_1|_\mathrm{min}</math> for all .

For reference purposes, Figure 2 has been displayed here, again, in the lefthand panel of Figure 4; the animation sequence presented in the righthand panel illustrates how the <math>\xi_1</math>-circle (depicted by the locus of small black dots) intersects the surface of the (pink) equatorial-plane torus as the value of <math>~\xi_1</math> is varied over the parameter range,
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<math>~\xi_1|_\mathrm{max} \geq \xi_1 \geq ~\xi_1|_\mathrm{min} \, ,</math>
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for a toroidal coordinate system whose origin (filled, red dot) remains fixed at the (cylindrical) coordinate location, <math>~(\varpi, z) = (a, Z_0) = (\tfrac{1}{3}, \tfrac{3}{4})</math>.  For a toroidal coordinate system with this specified origin and an equatorial-plane torus having <math>~\varpi_t = \tfrac{3}{4}</math> and <math>~r_t = \tfrac{1}{4}</math> &#8212; as recorded in the top row of numbers in the Table, below &#8212; the <math>\xi_1</math>-circle makes ''first contact'' with the torus when <math>~\xi_1 = \xi_1|_\mathrm{max} = 1.1927843</math> and it makes ''final contact'' when <math>~\xi_1 = \xi_1|_\mathrm{min} = 1.0449467</math>.  The animation sequence contains ten unique frames: The value of <math>~\xi_1</math> that is associated with the <math>\xi_1</math>-circle in each case appears near the bottom-right corner of the animation frame.  These parameter values have also been recorded in the first column of ten separate rows in the following table, along with other relevant parameter values.  For example, in each frame of the animation, the points of intersection between the surface of the torus and the <math>\xi_1</math>-circle are identified by filled, green diamonds; the (cylindrical) coordinates associated with these points of intersection, <math>~(\varpi_i, z_i)</math>, are listed in each table row, along with the corresponding value of the toroidal coordinate system's angular, <math>~\xi_2</math> coordinate.

Specific values for these parameters are tabulated in a [[Appendix/Ramblings/ToroidalCoordinates#Example2|Table titled, ''Example 2'']] in our accompanying notes.
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Notice in the animation that, while the origin of the selected toroidal coordinate system (the filled red dot) remains fixed, the ''center'' of the <math>\xi_1</math>-circle does not remain fixed.  In order to highlight this behavior, the location of the center of the <math>\xi_1</math>-circle has been marked by a filled, light-blue square and, in keeping with the [[#THH12Figure4|earlier Figure 1 diagram]], a vertical, light-blue line connects this center to the equatorial plane of the cylindrical coordinate system.